Double Yangians and quantum vertex algebras, I

For any symmetrizable generalized Cartan matrix $A$, we introduce an algebra $\widehat{\mathcal{D}\mathcal{Y}}(A)$, which is essentially the centrally extended double Yangian when $A$ is of finite type, and we give a new field (current) presentation of $\widehat{\mathcal{D}\mathcal{Y}}(A)$. Among the main results, for any $\mathrm{\ell }\in \mathbb{C}$ we construct a universal vacuum $\widehat{\mathcal{D}\mathcal{Y}}(A)$-module ${\mathcal{V}}_A(\mathrm{\ell })$ of level $\mathrm{\ell }$, prove that there exists a natural $\mathrm{\hslash }$-adic weak quantum vertex algebra structure on ${\mathcal{V}}_A(\mathrm{\ell })$, and give an isomorphism between the category of restricted $\widehat{\mathcal{D}\mathcal{Y}}(A)$-modules of level $\mathrm{\ell }$ and the category of ${\mathcal{V}}_A(\mathrm{\ell })$-modules.